Cycles: "Image as plane" show shadow from other objects
There are two solutions. Option 1 is easier and better.

Option 1 (Diffuse)

When importing the the image, make sure that you have Diffuse selected under Material Settings on the left side.



Option 2 (Emission)

I have searched far and wide for a solution to test for a shadow so that one can use the Emission setting with some extra nodes, but eventually concluded that it cannot be done. The final word came from gandalf3's answer here. You can, however, do a workaround through RenderLayers, if you so desire.

Final result:



Nodes:



What I have setup are two RenderLayers. One with the cube and image (Scene), and one with a plane with identical dimensions to the image with a plain Diffuse BSDF material (Shadow). I also enabled Transparent under Properties panel > Render tab > Film drop down.

This is the .blend file: